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Chapter 10: Problem 42

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,卤3)\(;\) asymptotes: \(y=\pm 3 x\)

Short Answer

Expert verified
The standard form of the equation of the hyperbola is \(\frac{y^2}{9} - \frac{x^2}{81} = 1\).

Step by step solution

01

Identifying a and b

A hyperbola that opens left and right has a standard form of \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), while one that opens up and down has a standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Since the vertices of the hyperbola are placed vertically at (0,卤3), we know that our hyperbola opens up and down. 'a' is the distance from the center to each vertex. Given that our vertices are (0,卤3), the value of 'a' is 3.
02

Find b from the asymptotes

The equation of the asymptotes of a hyperbola has the form y = 卤(b/a)x. In this case, the asymptotes are given as \(y=\pm 3x\), which indicates b/a = 3. We already have a = 3 from Step 1, so we solve this equation for b: \(3 = \frac{b}{3}\) which gives b = 9.
03

Write down the equation of the hyperbola

Now, we can write down equation for the hyperbola using the identified values of a and b. Plugging a = 3 and b = 9 into the standard form of the equation of the hyperbola gives \(\frac{y^2}{9} - \frac{x^2}{81} = 1\).

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